Gradient estimates for harmonic and generalized harmonic functions

We study $h^p_{(\alpha, \beta)}$ classes of $(\alpha, \beta)$ harmonic functions in the unit disc, which are analogous to the classical Hardy spaces of harmonic functions. Specifically, we obtain sharp estimates of $Du(0)$ for $u \in h^p_{\alpha, \beta)$ in terms of the $L^p$ norm of the boundary function. Asymptotically sharp estimates are obtained for $Du(z)$, as $|z| \to 1$, as well as for the higher order derivatives.
In the second part of the talk we investigate vertical derivative of a harmonic function in the upper half space, estimates are expressed in terms of the modulus of continuity of the boundary function.